Example: The car accident was my fault. I made the mistake of making a left turn on red.

Half answer: am I mistaken saying that mistake comes from: " missing a take " ? In this case it seems to imply an involuntary error. As I am never at any fault, the other word is completely new to me.

Well spotted Andrew! To me, a "mis-take" means you have taken the situation incorrectly. Though not actually 'missed' a take; the "mis-" prefix simply means "wrongly". "Fault" refers more closely to a deficiency or defect. If you want to stick close to the etymology. It's my mistake (= because I did not understand the situation properly) It's my fault (=due to a lack of ability/attention/etc)

A mistake is something that happens on accident, while fault is taking ownership of that mistake

A mistake is something that appears on the "outside". Generally, you can erase it, or clean it up. If you make a mistake in your test, you erase it and correct it. A fault is something "internal". If a machine isn't working, it has a fault. Or it is faulty.